DYNAMICS OF STRONGLY DAMPED WAVE EQUATIONS IN LOCALLY UNIFORM SPACES: ATTRACTORS AND ASYMPTOTIC REGULARITY

This paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces. After proving the global well-posedness, we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded (in H(lu)(2)(R(N)) x H(lu)(1)(R(N))) subset which attracts exponentially every initial H(lu)(1)(R(N)) x L(lu)(2)(R(N))-bounded set with respect to the H(lu)(1)(R(N)) x L(lu)(2)(R(N))-norm. Then, we show there is a (H(lu)(1)(R(N)) x L(lu)(2)(R(N)), H(p)(1)(R(N)) x H(p)(1)(R(N)))-global attractor, which reflects the strongly damped property of Delta(ut) to some extent.